{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "### 矩阵\n",
    "矩阵可以看成是行向量或者列向量的集合\n",
    "### 向量函数与矩阵函数\n",
    "作用域是向量空间的函数，我们称之为向量函数。矩阵函数是一类特殊的向量函数，其作用域和值域均为矩阵。\n",
    "\n",
    "\n",
    "### 系数矩阵和增广矩阵\n",
    "系数矩阵就是将方程组的系数提出来作为矩阵，增加矩阵是再系数矩阵的基础上增加值向量。\n",
    "\n",
    "### 方阵和零矩阵\n",
    "方阵的行列相等，零矩阵所有矩阵元素都为0.\n",
    "\n",
    "### 对角矩阵\n",
    "N阶方阵A，除对角线外，其余位置均为0.即为对角矩阵。记为$A=diag(\\lambda_1,...,\\lambda_n)$\n",
    "另有对角矩阵$B=diag(\\eta_1,...,\\eta_n)$,则A与B矩阵的乘积为对角矩阵$A\\times B=diag(\\lambda_1\\eta_1,...,\\lambda_n\\eta_n) $\n",
    "\n",
    "### 单元矩阵\n",
    "N阶对角矩阵上的对角线元素均为1，此次称为单位矩阵，即为E或者I。对于任意矩阵B，与单位阵的乘积\n",
    "\n",
    "### 矩阵的乘法\n",
    "矩阵的乘法$A\\cdot B$可以看成是A的行向量与B的列向量的的点积。\n",
    "\n",
    "### 矩阵的转置\n",
    "矩阵$A=(a_{ij})$的转置被定义为$A^T=(a_{ji})$。若矩阵$A=A^T$,则称A为对称矩阵，若$A=-A^T$,则称A为反对称矩阵。\n",
    "矩阵转置有如下性质：$(A\\cdot B)^T=B^T\\cdot A^T$,同理，$(A^N)^T=(A^T)^N$\n",
    "\n",
    "### 线性函数的定义\n",
    "若函数f(x)满足：$f(cx)=cf(x)\\\\f(x+y)=f(x)+f(y)$，则称f(x)是线性函数。矩阵函数是线性函数的一种。\n",
    "\n",
    "### 常用矩阵的几何意义\n",
    "对于列向量$x=(x1,x2)^T$,假定经过矩阵函数变换后得到列向量$y$,即$Ax=y$\n",
    "若A为单位矩阵，则x=y;\n",
    "\n",
    "若$A=\\begin{bmatrix}0&1\\\\1&0\\end{bmatrix}$，则$y=(x2,x1)^T$，相当于翻转,此时矩阵A称为镜像矩阵。\n",
    "\n",
    "若$A=\\left[\\begin{matrix}m&0\\\\0&n\\end{matrix}\\right]$，则$y=(mx1,nx2)^T$，相当于伸缩,此时矩阵A称为伸缩矩阵。\n",
    "\n",
    "若$A=\\left[\\begin{matrix}1&m\\\\n&1\\end{matrix}\\right]$，则$y=(x1+mx2,nx1+x2)^T$，相当于平移,此时矩阵A称为剪切矩阵。\n",
    "\n",
    "若$A=\\left[\\begin{matrix}\\cos\\theta&-\\sin\\theta\\\\\\sin\\theta&\\cos\\theta\\end{matrix}\\right]$，则$y=(\\cos\\theta x1-\\sin\\theta x2,\\sin\\theta x1+\\cos\\theta x2)^T$，相当于逆时针旋转$\\theta$,此时A称为旋转矩阵。"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}